Tweaking Results

In Chapter 5, we introduced tweaking and the mathematics behind our algorithm for aligning separated $\beta$ -strands. In this chapter, we will present examples of tweaking at work and how it affects the energetics of our protein folds. We will also gather some data regarding the efficacy of tweaking, the number of times tweaking is applied, and how successful those attempts are. We will then fold some random sequences utilizing a scoring function that promotes the production of $\beta$ -sheets. We will finish with a prediction for a short fifty-six amino acid protein, protein G (1PGB) [30].

As part of our effort to debug tweaking and generate a large number of tweaking instances, we added a debugging module to tweaking. This module generates proteins with short to medium length stretches of Gly. We restrict the Gly amino acids to only a single $\beta$ -sheet torsion angle pair ( $-139^\circ$ , $135^\circ$ ), and we then utilize a simplified scoring function that attempts to maximize the number of tweaking instances and the distance between the mating amino acids within a single protein. While this is not a good setup to solve for actual protein structures, it does demonstrate the capabilities of our tweaking work quite clearly. The examples from this chapter use this module unless otherwise noted.

Mis-aligned and Re-aligned Proteins

Tweaking is able to align both parallel and anti-parallel strands. We will select some examples that illustrate the variety of behaviors one can see in tweaking.

**Figure 7.1:** Examples of before and after alignment of parallel and anti-parallel $\beta$ -sheets created by tweaking.
$\begin{displaymath} \begin{array}{c c} \multicolumn{1}{l}{\mbox{\bf Anti-Paralle... ...ncludegraphics[height=2.5in]{GRAPHICS/tweak_1a.ps} \end{array}\end{displaymath}$

In Figure 7.1, we show a pair of successful tweaking events. The first is a short turn region that is tweaked so that the two $\beta$ -sheets are able to create an anti-parallel $\beta$ -sheet. The two amino acids aligned here are close to each other on the backbone (amino acids 8 and 12). In order to achieve this alignment, we tweaked nine backbone angles on three amino acids (9-11). The size of the changes are in Table 7.1.

Table 7.1: Change in backbone torsion angles necessary to align amino acids eight and twelve.

Angle	Tweaked	Original
$\phi_{9}$	-69.702194	-105.158663
$\psi_{9}$	+17.882301	-5.627044
$\omega_{9}$	+167.630625	+179.959705
$\phi_{10}$	-58.331865	-65.616084
$\psi_{10}$	-73.781114	-25.691042
$\omega_{10}$	+165.475892	-179.950169
$\phi_{11}$	-136.077522	-102.008904
$\psi_{11}$	-24.154507	+10.009309
$\omega_{11}$	-106.512036	-179.926125

On average, the three $\phi$ angles were tweaked $25.6^\circ$ , the $\psi$ angles $35.2^\circ$ , and the $\omega$ angles $33.4^\circ$ . Since we were being as permissive as possible in the construction of our proteins, we allowed the $\omega$ torsion angles to vary just as easily as the non- $\omega$ torsion angles. It is clear that a naturally occurring protein would not have an $\omega$ angle of $-106^\circ$ . It is a relatively simple matter to freeze those angles during a tweaking event.

The second example demonstrates a tweaking event where many of the amino acids are frozen. This tweaking event leads to a parallel $\beta$ -sheet. The two bonding amino acids (2 and 32) are far apart on the backbone. In order to align these two atoms we tweak a total of 27 backbone bond angles in 9 amino acids (out of a total of 29 possible). The untweaked amino acids are not changed because they are part of a $\beta$ -strand, and tweaking always respects existing secondary structure. We won't present all 27 angles changed here, but we will summarize the results. On average, the nine $\phi$ angles were tweaked $6.7^\circ$ , the $\psi$ angles $7.5^\circ$ , and the $\omega$ angles $6.4^\circ$ .

If you consider the figures presented (Figure 7.1), the anti-parallel case has much less slack in its alignment. This requires the tweaking algorithm to wrench the protein around into alignment. The parallel case has more opportunity to alter the intervening angles, so it is able to effect more subtle changes to bring to two strands into alignment.

$\begin{displaymath} \begin{array}{c c} \multicolumn{1}{l}{\mbox{\bf Tweaked \mbo... ...6}} & \multicolumn{1}{l}{\mbox{\bf Score: 75736}} \end{array}\end{displaymath}$

[A parallel $\beta$ -sheet with the score between tweaked and untweaked versions compared.] We seek to minimize our protein's score.

We can also use our scoring function to determine if these alignments are energetically advantageous for the protein. We would expect that they are as the alignment creates a greater number of hydrogen bonding and reduces the accessible surface area of the fully extended protein. To effectively measure this we have extended the second $\beta$ -strand two additional amino acids to create a $\beta$ -sheet from the now aligned $\beta$ -strands. As Figure 7.2 shows, the score of the tweaked protein is far better than that of the misaligned protein. This also points out why tweaking events are less likely to occur when the optimism (see Section 3.3) of our search is low. While we see a dramatic improvement from a tweaking event, the energy function will view the partial fold shown in Figure 7.2 poorly and would possibly prune that partial solution when optimism is low.

**Figure 7.3:** An example of a nested tweaking event.
$\begin{displaymath} \begin{array}{c c} \multicolumn{1}{l}{\mbox{\bf Un-Tweaked}}... ...includegraphics[height=2.5in]{GRAPHICS/tweak_4b.ps} \end{array}\end{displaymath}$

Figure 7.3 displays another capability of tweaking. In the figure, amino acids 17 and 73 are aligned. In between those two amino acids, we already have alignments between amino acids 2 and 32, 33 and 40, 43 and 52, and 58 and 63. Tweaking maintained each of these prior alignments and did not destroy pre-determined structure. This feature allows us to bring global $\beta$ -strands into alignment without threatening more local structures.

Using our specialized tweaking environment, we are able to generate protein folds with a large number of $\beta$ -sheets. While this ``solution'' (see Figure 7.4) does not represent an actual protein, it does show that $\beta$ -sheets are possible in the HOPS folding environment. When given the proper inducement, HOPS will locate a good deal of sheet behavior.

**Figure 7.4:** A solution to a randomly generated 80 amino acid protein generated by our tweaking environment.
$\includegraphics[width=3.0in]{GRAPHICS/tweak_soln.ps}$

Tweaking Opportunity

While the examples above show that tweaking can align two $\beta$ -strands, it does not imply that all attempts at alignment are successful. In fact, the vast majority of alignment attempts are unsuccessful. In most cases, the two atoms tweaking is working to align are too far apart from each other to align. If there is an insufficient number of tweakable amino acids between the two targets, there will be no way to bring the amino acids together even if they are relatively close to each other.⁴⁷

Even if we can determine a sequence of torsion angles that align the two targets, this contortion will often introduce steric clashes which disqualify the tweaked conformation from consideration (see Figure 7.5).

**Figure 7.5:** An example of a tweak leading to a clash.
$\includegraphics[width=3.0in]{GRAPHICS/tweak_clash.eps}$

We have run a series of tests of tests to determine what number of attempted tweaks successfully align the two strands and what percentage of those then survive without incurring an immediate steric clash. In the first test, we used our tweak-conducive environment and ran the algorithm ten times creating random proteins 20 amino acids in length. In the second test, we ran tweaking HOPS also 10 times creating random proteins 20 amino acids in length. However, this time we used standard torsion angles and did not use a scoring function that directly promotes the creation of $\beta$ -sheets. Instead, we used our standard scoring function.

In the first test, we attempted to construct a tweak 2,680,746 times, but were successful only 3,176 times ( $0.11\%$ of all attempts). Of those 3,176 successful alignments, only 711 did not create steric clashes. In the second test, we were much more successful. A total of 54,133,738 tweaks were attempted, and 1,453,486 were successful ( $2.7\%$ ). Of the successful tweaks 36,274 did not result in a steric clash. This means that approximately $0.1\%$ of all tweaks in a realistic situation were successful. While this may seem discouraging, a successful tweaking event does have a significant benefit energetically.

Predicted Structures with Sheets

Naturally, the goal of tweaking is the ab initio prediction of $\beta$ -sheet structure in real proteins. While we have established that tweaking will work under idealized conditions, we will now attempt to predict the structure of an already known structure, protein G. Protein G was mentioned earlier in Chapter 2. It is a small protein (56 amino acids) which has a pair of $\beta$ -sheets with a helix in between. The two anti-parallel $\beta$ -sheets then come together to form a larger sheet. This combination of sheet and helix structure makes it a quality target for our technique.

The standards of success in ab initio structure prediction are difficult to define. There are only a few ab initio predictions in each CASP that can be considered accurate predictions of a protein's structure. While we do not at the moment have a program that we would enter into blind tests such as CASP, we are moving in that direction. As an example of progress in this direction, we have run tests on protein G with a single caveat. For input torsion angles, we have utilized the actual angles found in the protein and a standard sheet angle. This does not tilt things as dramatically into our favor as it might seem. Only 13 of the 56 amino acids had between two and four angle choices in this test and there were 12 possible torsion angles for each of the 11 Thr amino acids. This guarantees that our discrete search space still contains well over $10^{39}$ possible conformations.

For the results presented here, we ran a test for 80 hours on a network of 8-10 Pentium computers (varying in speed from 1GHz to 75MHz). All computers utilize the Linux operating system. The best solution we gathered is not guaranteed to be the optimum as we did not run to completion. The results are still encouraging as HOPS correctly chose the torsion angles for the central helix and also created $\beta$ -strands and a pair of $\beta$ -sheets. HOPS and tweaking brought out a good deal of the sheet structure found in the native fold which contains a pair of anti-parallel $\beta$ -sheets that then fold together to form a parallel $\beta$ -sheet. We have the first anti-parallel $\beta$ -sheet, however, the bonding pattern is off slightly as we have matched amino acids 7 and 24, compared to 7 and 15 in the native fold. However, we match amino acids 5 and 42 in a parallel $\beta$ -sheet, compared to amino acids 6 and 52 in the native fold. A printout of the difference in torsion angles is presented in Appendix D. The actual and predicted structures are side-by-side in Figure 7.6.

**Figure 7.6:** An intermediate prediction by HOPS for protein G.
$\begin{displaymath} \begin{array}{c c c} \includegraphics[width=2.5in]{GRAPHICS/... ...lumn{1}{c}{\mbox{\bf Prediction Score: 6941 }} \\ \end{array}\end{displaymath}$